a = 8/5  b = 1  lambda = 64/169  a''^2 = 9216/4225  a'' = 96/65  b''^2 = 105/169  c^2 = 39/25

==== orbit I (vertex at (a,0)) ====
  P1 = (8/5, 0)
  P2 = (64/65, sqrt(105)/13)
  P3 = (-64/65, sqrt(105)/13)
  P4 = (-8/5, 0)
  P5 = (-64/65, -sqrt(105)/13)
  P6 = (64/65, -sqrt(105)/13)
  (1) all vertices on e: True
  (2) side P1 P2: tangent to caustic: True; contact (1152/845, 5*sqrt(105)/169) on caustic: True, on line: True, inside segment (s=5/13): True
  (2) side P2 P3: tangent to caustic: True; contact (0, sqrt(105)/13) on caustic: True, on line: True, inside segment (s=1/2): True
  (2) side P3 P4: tangent to caustic: True; contact (-1152/845, 5*sqrt(105)/169) on caustic: True, on line: True, inside segment (s=8/13): True
  (2) side P4 P5: tangent to caustic: True; contact (-1152/845, -5*sqrt(105)/169) on caustic: True, on line: True, inside segment (s=5/13): True
  (2) side P5 P6: tangent to caustic: True; contact (0, -sqrt(105)/13) on caustic: True, on line: True, inside segment (s=1/2): True
  (2) side P6 P1: tangent to caustic: True; contact (1152/845, -5*sqrt(105)/169) on caustic: True, on line: True, inside segment (s=8/13): True
  (3) reflection at P1: |u|=1 |v|=1  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P2: |u|=1 |v|=128/65  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P3: |u|=128/65 |v|=1  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P4: |u|=1 |v|=1  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P5: |u|=1 |v|=128/65  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P6: |u|=128/65 |v|=1  bisector x normal = 0: True (bisector nonzero: True)
  (4) 6 distinct vertices: True ; all turns positive (convex, ccw): True
  perimeter L = 516/65

==== orbit II (vertex at (0,b)) ====
  P1 = (0, 1)
  P2 = (-96/65, 5/13)
  P3 = (-96/65, -5/13)
  P4 = (0, -1)
  P5 = (96/65, -5/13)
  P6 = (96/65, 5/13)
  (1) all vertices on e: True
  (2) side P1 P2: tangent to caustic: True; contact (-768/845, 105/169) on caustic: True, on line: True, inside segment (s=8/13): True
  (2) side P2 P3: tangent to caustic: True; contact (-96/65, 0) on caustic: True, on line: True, inside segment (s=1/2): True
  (2) side P3 P4: tangent to caustic: True; contact (-768/845, -105/169) on caustic: True, on line: True, inside segment (s=5/13): True
  (2) side P4 P5: tangent to caustic: True; contact (768/845, -105/169) on caustic: True, on line: True, inside segment (s=8/13): True
  (2) side P5 P6: tangent to caustic: True; contact (96/65, 0) on caustic: True, on line: True, inside segment (s=1/2): True
  (2) side P6 P1: tangent to caustic: True; contact (768/845, 105/169) on caustic: True, on line: True, inside segment (s=5/13): True
  (3) reflection at P1: |u|=8/5 |v|=8/5  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P2: |u|=8/5 |v|=10/13  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P3: |u|=10/13 |v|=8/5  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P4: |u|=8/5 |v|=8/5  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P5: |u|=8/5 |v|=10/13  bisector x normal = 0: True (bisector nonzero: True)
  (3) reflection at P6: |u|=10/13 |v|=8/5  bisector x normal = 0: True (bisector nonzero: True)
  (4) 6 distinct vertices: True ; all turns positive (convex, ccw): True
  perimeter L = 516/65

  areas of orbit I: A = 336*sqrt(105)/845,  A' = 96*sqrt(105)/175,  A'' = 41472*sqrt(105)/142805
  M = O:
     antipedal of P  vertices: [(8/5, sqrt(105)/525), (0, 517*sqrt(105)/2625), (-8/5, sqrt(105)/525), (-8/5, -sqrt(105)/525), (0, -517*sqrt(105)/2625), (8/5, -sqrt(105)/525)]
     A*_M (antipedal of P) = 2784*sqrt(105)/4375   [numeric 6.5205739274129192912]
     A'*_M (antipedal of P') = 11323*sqrt(105)/14700   [numeric 7.8929403757116008499]
     A_M (pedal of P) = 43344*sqrt(105)/142805   A'_M (pedal of P') = 21*sqrt(105)/50
     k401  A' * A*_M      = 801792/21875   [numeric 36.653348571428571429]
     k402  A' / A*_M      = 25/29   [numeric 0.86206896551724137931]
     k403  A_M * A*_M     = 362009088/17850625   [numeric 20.279911095549875705]
     k404  A*_M / A_M     = 1656538/790125   [numeric 2.0965518114222433159]
     k204  A / A_M        = 169/129   [numeric 1.3100775193798449612]
     k303a A' * A'_M      = 3024/125   [numeric 24.192000000000000000]
     variants: A*A*_M = 2806272/105625, A''*A*_M = 346374144/17850625, A'*A'*_M = 271752/6125, A*A'*_M = 804/25, A''*A'*_M = 694656/29575
  M = f1:
     antipedal of P  vertices: [(8/5, (sqrt(105) + 24*sqrt(455))/525), (-sqrt(39)/5, 2*sqrt(105)/525), (-8/5, (-24*sqrt(455) + sqrt(105))/525), (-8/5, (-sqrt(105) + 24*sqrt(455))/525), (-sqrt(39)/5, -2*sqrt(105)/525), (8/5, (-24*sqrt(455) - sqrt(105))/525)]
     A*_M (antipedal of P) = 32*sqrt(105)/125   [numeric 2.6232193960856571861]
     A'*_M (antipedal of P') = 9152*sqrt(105)/18375   [numeric 5.1036785529285575185]
     A_M (pedal of P) = 56448*sqrt(105)/142805   A'_M (pedal of P') = 168*sqrt(105)/325
     k401  A' * A*_M      = 9216/625   [numeric 14.745600000000000000]
     k402  A' / A*_M      = 15/7   [numeric 2.1428571428571428571]
     k403  A_M * A*_M     = 37933056/3570125   [numeric 10.625133853856657680]
     k404  A*_M / A_M     = 28561/44100   [numeric 0.64764172335600907030]
     k204  A / A_M        = 169/168   [numeric 1.0059523809523809524]
     k303a A' * A'_M      = 48384/1625   [numeric 29.774769230769230769]
     variants: A*A*_M = 225792/21125, A''*A*_M = 27869184/3570125, A'*A'*_M = 878592/30625, A*A'*_M = 33792/1625, A''*A'*_M = 29196288/1922375
  M = f2:
     antipedal of P  vertices: [(8/5, (-24*sqrt(455) + sqrt(105))/525), (sqrt(39)/5, 2*sqrt(105)/525), (-8/5, (sqrt(105) + 24*sqrt(455))/525), (-8/5, (-24*sqrt(455) - sqrt(105))/525), (sqrt(39)/5, -2*sqrt(105)/525), (8/5, (-sqrt(105) + 24*sqrt(455))/525)]
     A*_M (antipedal of P) = 32*sqrt(105)/125   [numeric 2.6232193960856571861]
     A'*_M (antipedal of P') = 9152*sqrt(105)/18375   [numeric 5.1036785529285575185]
     A_M (pedal of P) = 56448*sqrt(105)/142805   A'_M (pedal of P') = 168*sqrt(105)/325
     k401  A' * A*_M      = 9216/625   [numeric 14.745600000000000000]
     k402  A' / A*_M      = 15/7   [numeric 2.1428571428571428571]
     k403  A_M * A*_M     = 37933056/3570125   [numeric 10.625133853856657680]
     k404  A*_M / A_M     = 28561/44100   [numeric 0.64764172335600907030]
     k204  A / A_M        = 169/168   [numeric 1.0059523809523809524]
     k303a A' * A'_M      = 48384/1625   [numeric 29.774769230769230769]
     variants: A*A*_M = 225792/21125, A''*A*_M = 27869184/3570125, A'*A'*_M = 878592/30625, A*A'*_M = 33792/1625, A''*A'*_M = 29196288/1922375

  areas of orbit II: A = 3456/845,  A' = 28/5,  A'' = 84672/28561
  M = O:
     antipedal of P  vertices: [(-79/60, 1), (-757/480, 0), (-79/60, -1), (79/60, -1), (757/480, 0), (79/60, 1)]
     A*_M (antipedal of P) = 463/80   [numeric 5.7875000000000000000]
     A'*_M (antipedal of P') = 9464/1125   [numeric 8.4124444444444444444]
     A_M (pedal of P) = 445824/142805   A'_M (pedal of P') = 108/25
     k401  A' * A*_M      = 3241/100   [numeric 32.410000000000000000]
     k402  A' / A*_M      = 448/463   [numeric 0.96760259179265658747]
     k403  A_M * A*_M     = 12901032/714025   [numeric 18.068039634466580302]
     k404  A*_M / A_M     = 13223743/7133184   [numeric 1.8538345569103502728]
     k204  A / A_M        = 169/129   [numeric 1.3100775193798449612]
     k303a A' * A'_M      = 3024/125   [numeric 24.192000000000000000]
     variants: A*A*_M = 100008/4225, A''*A*_M = 2450196/142805, A'*A'*_M = 264992/5625, A*A'*_M = 21504/625, A''*A'*_M = 526848/21125
  M = f1:
     antipedal of P  vertices: [(-32/35 - 13*sqrt(39)/105, 6/175 - 32*sqrt(39)/175), (-64/35 + sqrt(39)/21, 0), (-32/35 - 13*sqrt(39)/105, -6/175 + 32*sqrt(39)/175), (32/35 - 13*sqrt(39)/105, -32*sqrt(39)/175 - 6/175), (sqrt(39)/21 + 64/35, 0), (32/35 - 13*sqrt(39)/105, 6/175 + 32*sqrt(39)/175)]
     A*_M (antipedal of P) = 2304/875   [numeric 2.6331428571428571429]
     A'*_M (antipedal of P') = 1144/225   [numeric 5.0844444444444444444]
     A_M (pedal of P) = 580608/142805   A'_M (pedal of P') = 1728/325
     k401  A' * A*_M      = 9216/625   [numeric 14.745600000000000000]
     k402  A' / A*_M      = 1225/576   [numeric 2.1267361111111111111]
     k403  A_M * A*_M     = 191102976/17850625   [numeric 10.705674227092888904]
     k404  A*_M / A_M     = 28561/44100   [numeric 0.64764172335600907030]
     k204  A / A_M        = 169/168   [numeric 1.0059523809523809524]
     k303a A' * A'_M      = 48384/1625   [numeric 29.774769230769230769]
     variants: A*A*_M = 7962624/739375, A''*A*_M = 27869184/3570125, A'*A'*_M = 32032/1125, A*A'*_M = 33792/1625, A''*A'*_M = 827904/54925
  M = f2:
     antipedal of P  vertices: [(-32/35 + 13*sqrt(39)/105, 6/175 + 32*sqrt(39)/175), (-64/35 - sqrt(39)/21, 0), (-32/35 + 13*sqrt(39)/105, -32*sqrt(39)/175 - 6/175), (13*sqrt(39)/105 + 32/35, -6/175 + 32*sqrt(39)/175), (64/35 - sqrt(39)/21, 0), (13*sqrt(39)/105 + 32/35, 6/175 - 32*sqrt(39)/175)]
     A*_M (antipedal of P) = 2304/875   [numeric 2.6331428571428571429]
     A'*_M (antipedal of P') = 1144/225   [numeric 5.0844444444444444444]
     A_M (pedal of P) = 580608/142805   A'_M (pedal of P') = 1728/325
     k401  A' * A*_M      = 9216/625   [numeric 14.745600000000000000]
     k402  A' / A*_M      = 1225/576   [numeric 2.1267361111111111111]
     k403  A_M * A*_M     = 191102976/17850625   [numeric 10.705674227092888904]
     k404  A*_M / A_M     = 28561/44100   [numeric 0.64764172335600907030]
     k204  A / A_M        = 169/168   [numeric 1.0059523809523809524]
     k303a A' * A'_M      = 48384/1625   [numeric 29.774769230769230769]
     variants: A*A*_M = 7962624/739375, A''*A*_M = 27869184/3570125, A'*A'*_M = 32032/1125, A*A'*_M = 33792/1625, A''*A'*_M = 827904/54925

==== comparison of the two members (same caustic lambda = 64/169) ====
  M = O:  A'A*_M(orbit I) = 801792/21875,  A'A*_M(orbit II) = 3241/100,  difference = 371293/87500,  equal: False
  M = f1:  A'A*_M(orbit I) = 9216/625,  A'A*_M(orbit II) = 9216/625,  difference = 0,  equal: True
  M = f2:  A'A*_M(orbit I) = 9216/625,  A'A*_M(orbit II) = 9216/625,  difference = 0,  equal: True

k401 as printed (every M) is refuted at M = O iff the two values differ: True
